Experimental Mechanics

, Volume 6, Issue 1, pp 13–22

Optical phenomena in photoelastic models by the rotation of principal axes

Investigation shows that, using the matrix representation of the solution of the equations of photoelasticity, an adequate description of the complicated optical phenomena in three-dimensional photoelastic models can be obtained
  • Hillar K. Aben

Abstract

The basic equations of three-dimensional photoelasticity are derived in a form which is simpler than that of equations known previously. Using the matrix representation of the solution of these equations, it is also shown that when rotation of principal axes is present there always exist two perpendicular directions of polarizer by which the light emerging from the model is linearly polarized. These polarization directions of the incident and emergent light are named primary and secondary characteristic directions, respectively. The experimental determination of characteristic directions, as well as of the phase retardation, gives three equations on every light path to determine the stress components in a three-dimensional model. A general algorithm of the method of characteristic directions is presented, and its application by determination of stress in shells by normal and tangential incidence is described. A further extension of the method to the general axisymmetric problem has been suggested.

List of Symbols

A1,A2

components of electric vector of light after transformation [eq (4)]

a1,…,ak,b1,…,be,c1,…,cm,d1,…,dk

constants which determine the distribution of stress in an axisymmetric model

B1,B2

components of electric vector of light after transformation [eq 7]

B10,B20

components of the incident-light vector on arbitrary coordinate axes

B10o,B20o

components of the incident-light vector in primary characteristic directions

B1o,B2o

components of the emergent-light vector in secondary characteristic directions

c

velocity of light in vacuum

C

\(\frac{1}{{2k}}\frac{{\omega ^2 }}{{c^2 }}\)

C0,C1

photoelastic constants

C

CC0

D1,D2

components of electric-induction vector of light

E1,E2

components of electric vector of light

f1,…,f4

functions which determine the distribution of stress in an axisymmetric model

G(γ)

diagnonal unitary matrix [eq (20)]

k

ωN/c

N

index of refraction of the non-stressed medium

R

\(\frac{{2\varphi _0 }}{\Delta }\)

r

outer radius of a cylindrical shell; radial coordinate in an axisymmetric model

S

\(\sqrt {1 + R^2 } \)

Sj)

matrix of rotation [eq (19)]

t

thickness of the model

U

unitary matrix [eq (17)]

U1

unitary matrix which transforms the incident-light vector into the plane of symmetry

U1*

transpose ofU1

u′, v′

primary characteristic directions

u″, v″

secondary characteristic directions

x1,x2

principal directions at the point of entrance of light

x1,x2

principal directions at the point of emergence of light

x1,x2,z

rectangular coordinates

z1

z2/rt

α

angle between conjugate characteristic directions

α1, α2

angles which determine the primary and secondary characteristic directions

β

\(\frac{{3(1 - \mu ^2 )}}{{r^2 t^2 }}\)

characteristic phase retardation which corresponds to the matrixU

2γ1

characteristic phase retardation which corresponds to the matrixU1

Δ

C't1 − σ2)

Δ*

phase retardation determined by [eq (37)]

σij

Kronecker's tensor

ij

tensor of dielectric constant

j

principal components of the tensor of dielectric constant perpendicular to the wave normal

μ

Poisson's ratio

ξ, ζ, θ

parameters of the matrixU

σij

stress tensor

σ1, σ2

principal stresses perpendicular to the wave normal

11 − σ22)n, (σ12)n

membrane stress components in a shell

11 − σ22)b, (σ12)b

maximum bending stresses in a shell

σn

½C′(σ11 − σ22)n

σb

½C′(σ11 − σ22)b

σ2b

longitudinal bending stress in a cylindrical shell

σϑb

circumferential bending stress in a cylindrical shell

σϑn

circumferential membrane stress in a cylindrical shell

σ2n

longitudinal membrane stress in a cylindrical shell

σ2b°

max σ2b

σ

\({\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}C'\left( {\frac{t}{r}\sigma \vartheta _n - \sigma _{2b} ^\circ } \right)\)

σr, σϑ, σ22, τ2r

stress components in an axisymmetric model

τn

C′(σ12)n

τ2r

shearing stress in a cylindrical shell

τ1r°

max τ1r

τ

C′τ2r°

ϕ

angle of rotation of principal axes

ϑ0

total angle of rotation of principal axes

ψ

1/2SΔ

ω

frequency of vibration of light

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References

  1. 1.
    Ginsburg, V. L., “On the Influence of the Terrestrial Magnetic Field on the Reflection of Radio Waves from The Ionosphere,”Jnl. Phys.,7 (6),289–304 (1943).MathSciNetGoogle Scholar
  2. 2.
    Ginsburg, V. L., “On the Investigation of Stress by the Optical Method,” (in Russian),Zh. Tekh. Fiz.,14 (3),181–192 (1944).MathSciNetGoogle Scholar
  3. 3.
    O'Rourke, R. C., “Three-Dimensional Photoelasticity,”Jnl. Appl. Phys.,22 (7),872–878 (1951).MATHMathSciNetGoogle Scholar
  4. 4.
    Proshke, V. M., “On the Investigation of Stress in Three-Dimensional Models” (in Russian),The Method of Polarized Light for Measurement of Stress, Publ.,Acad. Sci. of the USSR, Moscow, 214–219 (1956).Google Scholar
  5. 5.
    Aben, H. K., “On the Investigation of Three-Dimensional Photoelastic Models,” (in Russian),Izv. Akad. Nauk SSSR, Mekh. i Mashinostr., (4),40–46 (1964).MATHGoogle Scholar
  6. 6.
    Mindlin, R. D., andGoodman, L. E., “The Optical Equations of Three-Dimensional Photoelasticity,”Jnl. Appl. Phys.,20 (1),89–97 (1949).Google Scholar
  7. 7.
    Neumann, F., “Die Gesetze der Doppelbrechung des Lichtes in komprimierten oder ungleichförmig erwärmten unkrystallinischen Körpern,” Abh. d. Kön. Akad. d. Wissenschaften zu Berlin, Pt. II (1841).Google Scholar
  8. 8.
    Kuske, A., “Die Gesetzmässigkeiten der Doppelbrechung,”Optik,19 (5),261–272 (1962).MathSciNetGoogle Scholar
  9. 9.
    Kuske, A., “Beiträge zur spannungsoptischen Untersuchung von Flächentragwerken,”Abh. d. Dtsch. Akad. d. Wissenschaften zu Berlin, Kl. Math., Phys. u. Tchn.,4,115–126 (1962).Google Scholar
  10. 10.
    Drucker, D. C., andMindlin, R. D., “Stress Analysis by Three-Dimensional Photoelastic Methods,”Jnl. Appl. Phys.,11 (11)724–732 (1940).Google Scholar
  11. 11.
    Mindlin, R. D., “An Extension of the Photoelastic Method of Stress Measurement to Plates in Transverse Bending,”Jnl. Appl. Mech.,8 (4),A-187 (1941).Google Scholar
  12. 12.
    Indenbom, V. L., “About the Stress on the Surface of Glass Specimens,” (in Russian),Zh. Tekh. Fiz.,26 (2),370–374 (1956).Google Scholar
  13. 13.
    Aben, H. K., “Application of the Method of Photoelasticity for the Analysis of Buckled Plates,” (in Russian),Izv. Akad. Nauk Estonian SSR, Ser. Tekh. i. Fiz.-Mat. Nauk,6 (1),28–40 (1957).Google Scholar
  14. 14.
    Masaki, Jun-ichi, “On the Problem of Rotation of Polarizing Axes in Stress Measurement of Shell-Type Glass-to-Metal Seals,” Proc. 2nd Japan Congr. on Testing Matls. (Kyoto, 1958), 189–191 (1959).Google Scholar
  15. 15.
    Kuske, A., “Einige neue spannungsoptische Verfahren,” Selected Papers on Stress Analysis (Conf., Delft 1959), 58–65 (1961).Google Scholar
  16. 16.
    Mylonas, C., andDrucker, D. C., “Twisting Stresses in Tape,”Experimental Mechancis,1 (7),23–32 (1961).Google Scholar
  17. 17.
    Lee, L. H. N., “Effects of Rotation of Principal Stresses on Photoelastic Retardation,”Ibid.,4 (10),306–312 (1964).CrossRefGoogle Scholar
  18. 18.
    Aben, H., “On the Experimental Determination of Parameters of Complex Optical Systems,” (in Russian),Izv. Akad. Nauk Estonian SSR, Ser. Fiz.-Mat. i Tekh. Nauk,13 (4),329–343 (1964).Google Scholar
  19. 19.
    Jones, R. C., “A New Calculus for the Treatment of Optical Systems, I,”Jnl. Opt. Soc. Amer.,31 (7),488–493 (1941).MATHGoogle Scholar
  20. 20.
    Kuske, A., “Einführung in die Spannungsoptik,”Wissenschaftliche Verlaggesellschaft M.B.H., Stuttgart (1959).Google Scholar
  21. 21.
    Poincaré, H., “Théorie Mathématique de la Lumière, II,” Paris (1892).Google Scholar
  22. 22.
    Hurwitz, H., andJones, R. C., “A New Calculus for the Treatment of Optical Systems, II,”Jnl. Opt. Soc. Amer.,31 (7),493–499 (1941).Google Scholar
  23. 23.
    Robert, A., andMme Guillemet, “Nouvelle méthode d'utilisation de la lumière diffusée en photoélasticimétrie à trois dimensions,”Rev. Franç. de Méc., (5/6),147–157 (1963).Google Scholar
  24. 24.
    Bokshtein, M. F., “Geometrical Analysis of the Polarized Light while it Passes a Three-Dimensional Stressed Model” (in Russian), The Method of Polarized Light for Measurement of Stress (Conf., Leningrad 1958), Leningrad Univ. Publ., 102–107 (1960).Google Scholar
  25. 25.
    Plechata, R., “Determination of Resultant Parameters in a Three-Dimensional Elastically Loaded Continuum,”Acta Technica ČSAV, No.5, 432–439 (1964).Google Scholar
  26. 26.
    Aben, H. K., “Photoelastic Phenomena by Uniform Rotation of Principal Axes,” (in Russian)Izv. Akad. Nauk SSSR, Mekh. i. Mashinostr., (3),141–147 (1962).Google Scholar
  27. 27.
    Aben, K. H., “A Nomogram for the Interpretation of the Photoelastic Phenomena by Uniform Rotation of Principal Axes” (in Russian),Izv. Akad. Nauk SSSR, Mekh. i Mashinostr., (6),174–175 (1963).Google Scholar
  28. 28.
    Aben, H. K., and Saks, E. G., “Optical Phenomena by Passing of Light Through Shells” (in Russian), The Method of Polarized Light for Measurement of Stress (Conf., Leningrad 1958), Leningrad Univ. Publ., 208–221 (1960).Google Scholar
  29. 29.
    Kayser, R., “Spannungsoptische Untersuchung von allgemeinen Flächentragwerken unter direkter Beobachtung,” Dissertation, T. H. Stuttgart (1964).Google Scholar
  30. 30.
    Kuske, A., “L'analyse des Phénomènes optiques en photoélasticité à trois dimensions par la méthode du cercle de j,”Rev. Franç. de Méc., 9, 49–58 (1964).Google Scholar
  31. 31.
    Aben, H. K., “On the Application of Photoelastic coatings by the Investigation of Shells” (in Russian),Izv. Akad. Nauk. SSSR, Mekh. i Mashinostr.,7 (6),106–111 (1964).Google Scholar
  32. 32.
    Riney, T. D., “Photoelastic Determination of the Residual Stress in the Dome of Electron Tube Envelopes,”Proc. SESA,15 (1),161–170 (1957).Google Scholar
  33. 33.
    Masaki, Jun-ichi, “A Simple Method for Measuring the Stress in Shell-Type Glass-to-Metal Seals,” Proc. 1st Japan. Congr. on Testing Matls. (Tokyo 1957), 159–163 (1958).Google Scholar
  34. 34.
    O'Rourke, R. C. andSaenz, A. W., “Quenching Stresses in Transparent Isotropic Media and the Photoelastic Method,”Quart. Appl. Math.,8 (3),303–311 (1950).MathSciNetGoogle Scholar
  35. 35.
    Drucker, D. C., andWoodward, W. B., “Interpretation of Photoelastic Transmission Patterns for a Three-Dimensional Model,”Jnl. Appl. Phys.,25 (4),510–512 (1954).Google Scholar
  36. 36.
    Aben, H. K., “Some Problems of Superposing of Two Birefringent Plates,” (in Russian),Optika i Spektr.,15 (5),682–689 (1963).Google Scholar

Copyright information

© Society for Experimental Mechanics, Inc. 1966

Authors and Affiliations

  • Hillar K. Aben
    • 1
  1. 1.Department of Applied Mathemtics and Mechanics, Institute of CyberneticsAcademy of Sciences of the Estonian SSRTallinnU.S.S.R., Estonia

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